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Here’s an Easier Way To Calculate Standard Deviation

deviation pic

So you just started a statistics unit, and you’ve learned about a little thing called standard deviation. And you might be thinking, “Why in the world do I have to do all these computations for standard deviation?  What a pain! What’s the standard deviation measure anyway?!?!”

The standard deviation (SD) measures how far away your data is from the average, on average.  This means if you have a big SD. your data is really spread out, and a small SD means your data is not so spread out.

FUN FACT: Another way to say standard deviation is “root mean square of deviations from average”.  To compute the SD, we do these operations in reverse: find the average, compute the deviations from average, square those, find their mean, then take the square root.  And we have to do all this crazy squaring and square rooting because we don’t want negative values.  Why not use absolute value instead you ask? Well, it’s because absolute values don’t play nice with calculus.

Let’s compute the SD of this set of data the hard way, then I’ll show you the easy(ier) way {17, 35, 28, 25, 20, 32, 25, 22, 28, 30}.  First add up the set, divide by the number of elements in the set to get the mean.  In our case, it’s 26.2.  Then fill out this chart:


Okay, now find the mean of the deviations squared… it’s 27.6.  Then take the square root of that… it’s 5.25.

But omg woah that’s kinda complicated.  But there’s hope!  There’s an easier way!  You can use this formula instead:


What’s this say you ask? First we square all the entries.  Then find the average of those.  Subtract the mean of the data set squared, then take the square root.  Easy peasy.


The average of the entries squared is 714, minus the mean squared (761.76) is 27.6… and again, the square root of that is 5.25.

So there you go Brightstormers, two different ways to compute the SD.  Find which way you like best, then crush your next stats exam! For more help, be sure to check out Brightstorm math!

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